Question

1. given that tkmh is a parallelogram with diagonal hk.

prove that \\( \overline { t k } \cong \overline { h m } \\) and \\( \overline { t h } \cong \overline { k m } \\)

statements	reasons
2) \\( t k \parallel h m \\) and \\( t h \parallel k m \\)	2)
3) \\( \angle t h k \cong \angle m k h \\) and \\( \angle t k h \cong \angle m h k \\)	3)
4) \\( h k \cong h k \\)	4)
5) \\( \triangle t h k \cong \triangle m k h \\)	5)
6) \\( \overline { t k } \cong \overline { m h } \\) and \\( \overline { t h } \cong \overline { m k } \\)	6)

Explanation:

Step1: Given

Given that \( TKMH \) is a parallelogram with diagonal \( HK \).

Step2: Property of parallelogram

In a parallelogram, opposite sides are parallel. So, \( TK\parallel HM \) and \( TH\parallel KM \).

Step3: Alternate - interior angles

If two parallel lines are cut by a transversal, then alternate - interior angles are congruent. Since \( TK\parallel HM \) and \( HK \) is a transversal, \( \angle THK\cong\angle MKH \); since \( TH\parallel KM \) and \( HK \) is a transversal, \( \angle TKH\cong\angle MHK \).

Step4: Reflexive property

For any segment \( AB \), \( AB\cong AB \). So, \( HK\cong HK \).

Step5: ASA (Angle - Side - Angle) congruence

In \( \triangle THK \) and \( \triangle MKH \), we have \( \angle THK\cong\angle MKH \), \( HK\cong HK \), \( \angle TKH\cong\angle MHK \). By the ASA (Angle - Side - Angle) congruence criterion, \( \triangle THK\cong\triangle MKH \).

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle THK\cong\triangle MKH \), then \( TK\cong MH \) and \( TH\cong MK \) (by CPCTC).

Answer:

1. Given; 2) Property of parallelogram; 3) Alternate - interior angles theorem; 4) Reflexive property of congruence; 5) ASA (Angle - Side - Angle) congruence; 6) CPCTC (Corresponding Parts of Congruent Triangles are Congruent)